896 21 The Electronic Structure of Polyatomic Molecules
21.43List the symmetry operations that belong to benzene in its
equilibrium nuclear conformation. Specify the symmetry
operators of which each of the Hückel orbitals is an
eigenfunction and give the corresponding eigenvalues.
21.44a.List the symmetry operators that belong to the N 2 O
molecule if it has the structure NON.
b.List the symmetry operators that belong
to the N 2 O molecule if it has the structure
NNO.
21.45Consider a tennis ball or a baseball, ignoring any
trademarks or logos. List the symmetry operators that
belong to this object.21.9 Groups of Symmetry Operators
For many molecules, the equilibrium nuclear conformations have useful symmetry
properties. In Section 20.1 we introduced several symmetry operators: The identity
operator, the inversion operator, reflection operators, and rotation operators. We now
addimproper rotations, which are equivalent to ordinary rotations followed by a reflec-
tion through a plane perpendicular to the axis of rotation. The rotation axis is the sym-
metry element, and the axis and the reflection plane both include the origin. As with
ordinary rotations, we consider only counterclockwise rotations such that an integral
number of applications of the rotation yield a complete rotation of 360◦. An operator
that produces a full rotation withnoperations is denoted bŷSn. For example, an̂S 4
operator with thezaxis as its symmetry element will rotate by 90◦about thezaxis and
reflect through thexyplane:̂S 4 z(x,y,z)(x′,y′,z′)(−y,x,−z) (21.9-1)Sometimes the presence of one symmetry operator implies the presence of another. If
a molecule has anS 4 axis, it also has aC 2 axis, since the square of thêS 4 zoperator is
the same as thêC 2 zoperator:̂S 4 z^2 (x,y,z)̂C (^2) z(x,y,z)(−x,−y,z) (21.9-2)
Ifnis an even integer,
̂Snn̂E (neven) (21.9-3)
Ifnis an odd integer
̂Snn̂σ (nodd) (21.9-4)
where thêσoperator has the same reflection plane as thêSnoperator. An̂S 1 operation
has the same effect as the reflection operation, and an̂S 2 operation is equivalent to
the inversion operator̂i. The lowest-order improper rotation that is not equivalent to
another operator iŝS 3.
EXAMPLE21.13
Find the location to which the point (x,y,z) is moved by thêS 4 operator around theyaxis.
Solution
The rotation moves the point from (x,y,z)to(z,y,−x). The reflection through thexzplane
moves the point to (z,−y,−x).